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Threshold condensation to singular support for a Riesz equilibrium problem

Djalil Chafa\"iEdward B. SaffRobert S. Womersley
Jun 2022
摘要
We compute the equilibrium measure in dimension d=s+4 associated to a Rieszs-kernel interaction with an external field given by a power of the Euclideannorm. Our study reveals that the equilibrium measure can be a mixture of acontinuous part and a singular part. Depending on the value of the power, athreshold phenomenon occurs and consists of a dimension reduction orcondensation on the singular part. In particular, in the logarithmic case s=0(d=4), there is condensation on a sphere of special radius when the power ofthe external field becomes quadratic. This contrasts with the case d=s+3studied previously, which showed that the equilibrium measure is fullydimensional and supported on a ball. Our approach makes use, among other tools,of the Frostman or Euler-Lagrange variational characterization, the Funk-Heckeformula, the Gegenbauer orthogonal polynomials, and hypergeometric specialfunctions.
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